Advances in Materials Science and Engineering

Volume 2016, Article ID 9435431, 17 pages

http://dx.doi.org/10.1155/2016/9435431

## Modeling of Point Defects Annihilation in Multilayered Cu/Nb Composites under Irradiation

^{1}Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Università degli Studi di Cagliari, Via Marengo 2, 09123 Cagliari, Italy^{2}Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK

Received 15 July 2015; Accepted 16 December 2015

Academic Editor: Markku Leskela

Copyright © 2016 Sarah Fadda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work focuses on a mathematical modeling of the response to irradiation of a multilayer composite material. Nonstationary balance equations are utilized to account for production, recombination, transport, and annihilation, or removal, of vacancies and interstitials at interfaces. Although the model developed has general validity, Cu/Nb multilayers are used as case study. Layer thickness, temperature, radiation intensity, and surface recombination coefficients were varied systematically to investigate their effect on point defect annihilation processes at interfaces. It is shown that point defect annihilation at interfaces mostly depends on point defect diffusion. The ability of interfaces to remove point defects can be described by a simple map constructed using only two dimensionless parameters, which provides a general tool to estimate the efficiency of vacancy and interstitial removal in multilayer composite materials.

#### 1. Introduction

Intense irradiation, high temperature, high mechanical stresses, and the presence of chemically aggressive fluids make nuclear environment extreme [1, 2]. Under these conditions, materials undergo significant degradation processes that definitely limit their reliability, lifetime, and efficiency [2–10].

Point defect supersaturation due to ballistic atomic displacements induced by irradiation and the consequent generation of sustained net fluxes of vacancies and interstitials play a major role in the progressive damaging of the material [11, 12]. Biased production and annihilation of point defects depress recombination processes, allowing them time for clustering and nucleating voids [12]. Furthermore, interaction of vacancies with He atoms generated by neutron-activated transmutation reactions finally results in the precipitation of He bubbles [4, 9, 13]. These contribute to material embrittlement and hardening, typically accompanied by dimensional and chemical instabilities as well as by swelling, solute redistribution, creep, and associated processes [2, 4–7, 11–23].

Contrasting degradation processes and eventually enhancing the radiation tolerance of materials represent a stringent need in view of the increasing energy demand worldwide and the unsatisfactory development degree attained by massive energy production related to renewable sources. At the same time, improving radiation resistance poses a tremendous challenge to materials scientists and technologists addressing future fission and fusion reactors [11, 24–29].

In principle, a high density of unbiased radiation-induced point defect sinks could absorb, and annihilate, vacancy and interstitials by facilitating the recombination of Frenkel pairs [6, 10, 12, 13, 22, 30, 31]. In this respect, surfaces, grain boundaries, and interphase boundaries have attracted most interest [6, 21, 27, 32–37]. Due to the characteristic high densities of grain boundaries, nanostructured materials have recently gained much attention [10, 12, 20, 22, 23, 28, 38, 39]. Excess free volume and higher diffusivity of defects at grain boundaries, combined with shorter diffusion distances of point defects to interfaces, enhance, indeed, recombination probabilities [2, 4, 34, 37]. However, the use of nanostructured materials requires maintaining a high density of interfaces and grain boundaries during prolonged irradiation [39], which is not an easy task due to the thermal instability of nanocrystalline grains and their tendency to coarsen rapidly even at modest temperatures [28, 40, 41].

In this regard, a possible engineering strategy involves using multiphase nanostructured materials [6, 22, 42]. In particular, multilayer composites can be designed to exhibit enhanced radiation tolerance [6, 11, 20, 42–45]. For example, Cu/Nb nanolayered composites have been shown to possess satisfactory dimensional, chemical, and microstructural stability over a wide range of irradiation conditions [6, 7, 11, 37, 42, 46]. This is mostly due to the positive heat of mixing of Cu and Nb, which exhibit only terminal solid solubility and no tendency to form intermetallic compounds [11, 28].

The mechanisms underlying point defect annihilation at Cu/Nb interfaces have been studied recently by density functional theory (DFT) [47], molecular dynamics [1, 11, 21, 27, 34, 35, 48], and phase field calculations [37]. This provided considerable detail on the atomistic processes taking place on relatively short time scales. Nevertheless, the overall mechanistic scenario remains largely unknown [23]. Within this context, developing a continuum approach to describe long-term evolution of point defects under irradiation conditions is highly desirable.

Aimed at providing a contribution along this line, the present work concerns the development and validation of a mathematical model in space and time* continuum* addressing the dynamic behavior of vacancies and interstitials during irradiation. Nanometer-sized Cu/Nb multilayer composites are used as case study. Interfaces are described as a continuum spatial distribution of either neutral or variable-biased sinks [49]. This allows modeling both noncoherent interfaces such as grain boundaries and incoherent precipitates, which typically behave like neutral sinks, and coherent and semicoherent interfaces, which behave like variable-biased sinks [49]. Nonstationary balance equations for vacancy and interstitials are used to describe production, recombination, transport, and annihilation, or removal, of point defects at interfaces. Layer thickness, temperature, radiation conditions, and surface recombination coefficients were varied systematically to investigate their effect on annihilation processes at interfaces.

The numerical findings show that the kinetics of point defect annihilation at interfaces mostly depend on their diffusion into the material volume, whereas boundary structure plays a minor role. Furthermore, interfaces exhibit a capability of point defect removal amenable to simple description based on a map constructed using two parameters, namely, dimensionless vacancy diffusivity and the ratio between the characteristic times of point defect recombination and production. Such maps provide a general tool to estimate the efficiency of vacancy and interstitial removal and can be utilized for investigating the behavior of any given multilayer composite material.

#### 2. Mathematical Model

In this work, it is assumed that interfaces between materials are capable of adsorbing point defects. It is also assumed that there are no crossings of defects between interfaces and that distribution of defects on the interfaces is not spatially dependent. On the other hand, defect concentrations inside the layers are assumed to be dependent only upon the distance from the interfaces. Under these hypotheses, multilayer composite systems may be analyzed in one spatial coordinate adopting a simple reaction-diffusion model of a single layer bounded by two interfaces. The latter ones are assumed in this work to be perfectly stable under irradiation. This assumption means that intermixing is negligible and thus that the proposed model applies to those systems whose alternate layers are constituted by mutually immiscible elements.

The model takes into account the formation of Frenkel pairs at a constant and spatially uniform rate, as well as their diffusion and mutual annihilation. Radiation-induced vacancies and self-interstitial atoms are considered isolated; thus the formation of defect clusters like voids and dislocation loops is not taken into account. This simplifying assumption typically holds at low point defect concentrations because of the negligible cluster nucleation rate in these conditions. In addition, point defects annihilation at grain boundaries is also neglected. Interfaces are then the only sink for point defects taken into account in this model. This hypothesis is justified by experimental evidences in several multilayer composites [36]. In this regard, it should be pointed out that since clusters and grain boundaries act as sinks for point defects, the results of this model may be considered conservative in terms of point defect concentrations. In other words, the model outputs represent the maximum achievable point defect concentrations under the specified operating conditions.

The evolution of point defects concentration, that is, vacancies () and self-interstitial atoms (SIA) (), is described by the following one-dimensional spatial reaction-diffusion equation, along with its initial conditionwhere is the concentration of the point defect of type and is the diffusion coefficient of the point defect of type , while and are the production and the recombination rates of Frenkel pairs per unit volume, respectively. The reader should refer to Nomenclature for the significance of the other symbols.

Diffusion coefficients are expressed in terms of Arrhenius form for thermally activated events as follows:The production rate of Frenkel pairs has been calculated using the Stopping and Range of Ions in Matter (SRIM) technique and depends on radiation conditions and layer material lattice. The recombination rate of point defects is expressed as a second-order reaction:where the kinetic constant is given by is the concentration of the point defect of type at thermodynamic equilibrium, that is, the concentration value at a given temperature in absence of radiation. It depends on temperature according to the following equation:

Boundary conditions for (1) should take into account the characteristics of interfaces. In particular, if we assumed a planar surface, in the case of neutral sinks, the boundary conditions may be expressed as [50]where is the thickness of the layer. Under the assumption that the lattice is not severely distorted over the final jump region, the transfer velocity is equal towhere is the metal lattice spacing. It is worth pointing out that the boundary condition of a neutral sink can be alternately expressed aswhich coincides with the boundary condition of the so-called perfect sink.

In the case of interfaces acting as variable-biased sinks, boundaries are modeled considering a surface (interface) concentration of traps for vacancies, , and a surface (interface) concentration of traps for interstitials, . The occupation probability of traps of each type is taken to be and , respectively. An interstitial atom adjacent to the interface is assumed to be able to enter an unoccupied interstitial trap site or to recombine with the nearest neighbor trapped vacancy, jumping there from possible adjacent sites in the matrix. Similar processes are possible for vacancies. Moreover, trapped interstitials and vacancies may recombine on interfaces. According to this picture, boundary conditions at interfaces acting as a variable-biased sink may be expressed as [50]where traps occupation probabilities are obtained by solving the following steady-state balance equations [50]:

The parameter is the surface recombination coefficient that depends on the specific features of interfaces structure, such as the distance between trapping sites and their energy. It may be assumed that all the interface characteristics that affect point defect absorption or annihilation may be lumped in this parameter. Factor represents the number of different jumps to a site by which recombination can occur [50]. In this work, it is set to be equal to 4 for any material structure. In this way, the model gives the highest importance to the effect of the recombination between trapped point defects with respect to the recombination between a trapped point defect and a point defect jumping into the interface from the matrix (cf. (11)). Further details about this choice will be given in Section 3.

The model consisting of the balance equations (1) along with their initial conditions, (2), and their boundary conditions, that is, (7), in the case of interfaces acting as neutral sinks, or (10), in the case of interfaces acting as variable-biased sinks, allows one to describe the temporal evolution of point defect concentrations inside a single layer of a given material undergoing radiation.

A change of variables was used in this work in order to obtain dimensionless and normalized equations and parameters. This may help in analyzing the behavior of the system for different values of the parameters because this strategy typically reduces the number of parameters. Following the approach of Krantz [51], it is possible to define the dimensionless point defect concentrations aswhere superscripts and represent the reference and the scaling value, respectively. Accordingly, the dimensionless spatial coordinate and time may be defined as follows:

Let us now assume for the reference and scaling variables the following expressions:It may be worth noting that can be also regarded as the characteristic time of interstitials diffusion along the metal layer.

According to this change of variables, the evolution of point defects dimensionless concentrations as a function of the dimensionless time is described by the dimensionless equation,along with the initial conditionsThe dimensionless diffusion coefficient is defined with respect to the diffusion coefficient of SIAs:Consequently, the dimensionless diffusion coefficient for interstitials is equal to 1, while the dimensionless diffusion coefficient for vacancies is typically much smaller than 1, as the diffusion of interstitials is significantly faster than the diffusion of vacancies. The dimensionless parameter is expressed asand it represents the ratio between the characteristic times of point defect recombination and production phenomena.

Dimensionless boundary conditions for the case of neutral sinks are expressed aswhere the dimensionless parameter is defined as In the case of interfaces acting as variable-biased sinks, the boundary conditions may be expressed asThe dimensionless steady-state balance equations of occupation probabilities appear as follows:where the dimensionless parameter is defined asand expresses the ratio between the characteristic times of point defect annihilation at interfaces and net production phenomena.

With the aim of better evaluating metallic multilayers performances in terms of radiation resistance the following definition of efficiency is introduced in this work:It may be regarded as the efficiency of multilayers for -type point defects absorption or removal. The limiting concentrations and represent the minimum and maximum point defects concentrations achievable in metallic layers during irradiation, respectively.

While the minimum concentration may be easily identified as the equilibrium concentration consistently with (15), the maximum concentration deserves additional comments. Specifically, it may be thought of as the point defect concentration achieved under the particular condition of absence of sinks for point defects. With the aim of better clarifying this important point, let us consider a single layer of thickness bounded by two interfaces unable to absorb defects. Equations (1) still represent the balance equations for point defects within the layer. However, the coupled boundary conditions should be now expressed as insulated ones:Numerical solution of (1) along with boundary condition (31) showed that the maximum achievable point defect concentration resembles the stationary state one. In particular, it may be easily found thatwhich expresses the equilibrium between the recombination rate and the production rate. It may be easily recognized that the difference between maximum and minimum concentrations represents the scaling concentration introduced by (16), which is time-independent and assumes the same value for both vacancies and SIAs. It is worth noting that, according to its definition, efficiency measures the additional contribution of interfaces to point defect annihilation being the base case represented by the contribution of volumetric recombination only.

Eventually, according to the nondimensionalizing procedure, the efficiency of interfaces for point defects absorption may be also expressed asThe nondimensional model obtained through this change of variables has only four dimensionless parameters: , , , and . This model has been solved for a wide range of values of all the dimensionless parameters, paying attention to keeping their physical meaning.

The equations of the dimensionless model were solved numerically as a time-dependent problem. In the case of interfaces acting like variable-biased sinks, the system of equations consists of coupled algebraic and differential equations. The model was solved through Comsol Multiphysics using the diffusion and convection module and the ODE and DAE interfaces. The parameter dependences were studied through the parametric sweep extension step. The model parameters were those pertaining to copper and niobium and their values are reported in Tables 1 and 2, respectively.